Does group $A_6$ contain subgroup isomorphic with $S_4$ ?
The only thing that I ask for is any clue.
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2Find an element of order 4 in $A_6$, and a suitable element of order 3 in $A_6$, and see if they don't generate a copy of $S_4$. – Gerry Myerson Sep 04 '14 at 10:11
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4More generally, $A_{n+2}$ has a subgroup isomorphic to $S_n$. – Derek Holt Sep 04 '14 at 11:13
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This otherwise good question has been handled here at least twice. First here and more recently here. – Jyrki Lahtonen Sep 04 '14 at 14:01
2 Answers
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Hint: Take $S_4$ on the first four elements of the set permuted by $A_6$. Some of these are odd permutations. Can you see an easy way to convert them into even permutations which are contained in $A_6$? Can you do this in a consistent way which doesn't disturb the $S_4$ structure?
Mark Bennet
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Ok, slowly, it is difficult for me. Let $G$ will be subgroup of $A_6$ isomorphic to $S_4$. Thus, |G| = $S_4$. But I still have a problem. – xawey Sep 05 '14 at 12:25
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Going by Mark's answer's hint, you can prove that in general $\;S_n\;$ is embeddable in $\;A_{n+2}\;$ . Can you find an example when it is not enough to take $\;A_{n+1}\;$ instead?
Timbuc
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